3.2549 \(\int \frac{(5-x) (3+2 x)^{7/2}}{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=94 \[ -\frac{2}{21} (2 x+3)^{7/2}+\frac{62}{45} (2 x+3)^{5/2}+\frac{526}{81} (2 x+3)^{3/2}+\frac{3278}{81} \sqrt{2 x+3}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{4250}{81} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

(3278*Sqrt[3 + 2*x])/81 + (526*(3 + 2*x)^(3/2))/81 + (62*(3 + 2*x)^(5/2))/45 - (
2*(3 + 2*x)^(7/2))/21 + 12*ArcTanh[Sqrt[3 + 2*x]] - (4250*Sqrt[5/3]*ArcTanh[Sqrt
[3/5]*Sqrt[3 + 2*x]])/81

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Rubi [A]  time = 0.28198, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{2}{21} (2 x+3)^{7/2}+\frac{62}{45} (2 x+3)^{5/2}+\frac{526}{81} (2 x+3)^{3/2}+\frac{3278}{81} \sqrt{2 x+3}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{4250}{81} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(3 + 2*x)^(7/2))/(2 + 5*x + 3*x^2),x]

[Out]

(3278*Sqrt[3 + 2*x])/81 + (526*(3 + 2*x)^(3/2))/81 + (62*(3 + 2*x)^(5/2))/45 - (
2*(3 + 2*x)^(7/2))/21 + 12*ArcTanh[Sqrt[3 + 2*x]] - (4250*Sqrt[5/3]*ArcTanh[Sqrt
[3/5]*Sqrt[3 + 2*x]])/81

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Rubi in Sympy [A]  time = 48.8042, size = 83, normalized size = 0.88 \[ - \frac{2 \left (2 x + 3\right )^{\frac{7}{2}}}{21} + \frac{62 \left (2 x + 3\right )^{\frac{5}{2}}}{45} + \frac{526 \left (2 x + 3\right )^{\frac{3}{2}}}{81} + \frac{3278 \sqrt{2 x + 3}}{81} - \frac{4250 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{243} + 12 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**(7/2)/(3*x**2+5*x+2),x)

[Out]

-2*(2*x + 3)**(7/2)/21 + 62*(2*x + 3)**(5/2)/45 + 526*(2*x + 3)**(3/2)/81 + 3278
*sqrt(2*x + 3)/81 - 4250*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/243 + 12*atanh
(sqrt(2*x + 3))

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Mathematica [A]  time = 0.0864213, size = 112, normalized size = 1.19 \[ -\frac{2}{21} (2 x+3)^{7/2}+\frac{62}{45} (2 x+3)^{5/2}+\frac{526}{81} (2 x+3)^{3/2}+\frac{3278}{81} \sqrt{2 x+3}-6 \log \left (1-\sqrt{2 x+3}\right )+6 \log \left (\sqrt{2 x+3}+1\right )-\frac{4250}{81} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(3 + 2*x)^(7/2))/(2 + 5*x + 3*x^2),x]

[Out]

(3278*Sqrt[3 + 2*x])/81 + (526*(3 + 2*x)^(3/2))/81 + (62*(3 + 2*x)^(5/2))/45 - (
2*(3 + 2*x)^(7/2))/21 - (4250*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/81 - 6
*Log[1 - Sqrt[3 + 2*x]] + 6*Log[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.017, size = 80, normalized size = 0.9 \[ -{\frac{2}{21} \left ( 3+2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{62}{45} \left ( 3+2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{526}{81} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{3278}{81}\sqrt{3+2\,x}}-6\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{4250\,\sqrt{15}}{243}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }+6\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^(7/2)/(3*x^2+5*x+2),x)

[Out]

-2/21*(3+2*x)^(7/2)+62/45*(3+2*x)^(5/2)+526/81*(3+2*x)^(3/2)+3278/81*(3+2*x)^(1/
2)-6*ln(-1+(3+2*x)^(1/2))-4250/243*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)+
6*ln(1+(3+2*x)^(1/2))

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Maxima [A]  time = 0.798576, size = 131, normalized size = 1.39 \[ -\frac{2}{21} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{62}{45} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{526}{81} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{2125}{243} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{3278}{81} \, \sqrt{2 \, x + 3} + 6 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^(7/2)*(x - 5)/(3*x^2 + 5*x + 2),x, algorithm="maxima")

[Out]

-2/21*(2*x + 3)^(7/2) + 62/45*(2*x + 3)^(5/2) + 526/81*(2*x + 3)^(3/2) + 2125/24
3*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) + 327
8/81*sqrt(2*x + 3) + 6*log(sqrt(2*x + 3) + 1) - 6*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.289831, size = 134, normalized size = 1.43 \[ -\frac{1}{8505} \, \sqrt{3}{\left (8 \, \sqrt{3}{\left (270 \, x^{3} - 738 \, x^{2} - 8639 \, x - 24728\right )} \sqrt{2 \, x + 3} - 17010 \, \sqrt{3} \log \left (\sqrt{2 \, x + 3} + 1\right ) + 17010 \, \sqrt{3} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 74375 \, \sqrt{5} \log \left (\frac{\sqrt{3}{\left (3 \, x + 7\right )} - 3 \, \sqrt{5} \sqrt{2 \, x + 3}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^(7/2)*(x - 5)/(3*x^2 + 5*x + 2),x, algorithm="fricas")

[Out]

-1/8505*sqrt(3)*(8*sqrt(3)*(270*x^3 - 738*x^2 - 8639*x - 24728)*sqrt(2*x + 3) -
17010*sqrt(3)*log(sqrt(2*x + 3) + 1) + 17010*sqrt(3)*log(sqrt(2*x + 3) - 1) - 74
375*sqrt(5)*log((sqrt(3)*(3*x + 7) - 3*sqrt(5)*sqrt(2*x + 3))/(3*x + 2)))

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Sympy [A]  time = 138.764, size = 138, normalized size = 1.47 \[ - \frac{2 \left (2 x + 3\right )^{\frac{7}{2}}}{21} + \frac{62 \left (2 x + 3\right )^{\frac{5}{2}}}{45} + \frac{526 \left (2 x + 3\right )^{\frac{3}{2}}}{81} + \frac{3278 \sqrt{2 x + 3}}{81} + \frac{21250 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 < \frac{5}{3} \end{cases}\right )}{81} - 6 \log{\left (\sqrt{2 x + 3} - 1 \right )} + 6 \log{\left (\sqrt{2 x + 3} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**(7/2)/(3*x**2+5*x+2),x)

[Out]

-2*(2*x + 3)**(7/2)/21 + 62*(2*x + 3)**(5/2)/45 + 526*(2*x + 3)**(3/2)/81 + 3278
*sqrt(2*x + 3)/81 + 21250*Piecewise((-sqrt(15)*acoth(sqrt(15)*sqrt(2*x + 3)/5)/1
5, 2*x + 3 > 5/3), (-sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/15, 2*x + 3 < 5/3)
)/81 - 6*log(sqrt(2*x + 3) - 1) + 6*log(sqrt(2*x + 3) + 1)

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GIAC/XCAS [A]  time = 0.275098, size = 136, normalized size = 1.45 \[ -\frac{2}{21} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{62}{45} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{526}{81} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{2125}{243} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + \frac{3278}{81} \, \sqrt{2 \, x + 3} + 6 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 6 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^(7/2)*(x - 5)/(3*x^2 + 5*x + 2),x, algorithm="giac")

[Out]

-2/21*(2*x + 3)^(7/2) + 62/45*(2*x + 3)^(5/2) + 526/81*(2*x + 3)^(3/2) + 2125/24
3*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3)
)) + 3278/81*sqrt(2*x + 3) + 6*ln(sqrt(2*x + 3) + 1) - 6*ln(abs(sqrt(2*x + 3) -
1))